Oh my, I see some things happening here that you may want to remember. If a girl asks to hang out somewhere beside school, take her up on that. Just see where things go. Asking right off the bat whether they like you or not isn't going to get you very many places.
Although, it sounds like she just wants to be friends. I would still take her up on it. It's possible she likes you, but I wouldn't worry about it. It's high school, and most people in there are confused out of their minds. Just have her come over for some video games, or a movie, and leave it at that. She may just enjoy having someone to play games with and watch movies together.
You really need to quit thinking so much about it. Like, when she was heart broken, if you weren't thinking so much, you would have instinctively acted, by either comforting her, or going after him. Either would have won you some points in her book. This is the thing about relationships. You have to act, and put yourself out there. It's not always logical in how it works out. You need to understand that.
To be honest, now that you've given such a response, she may be a bit off-put to the idea of movies. If you let her know how you reacted to the whole scene, and how much it got to you, you might be ok. I would just tell her how seeing that happen made you feel, what you thought, and ask if she wants to come over. Just avoid pushing for a relationship at this point. The thing about AS is we may be a bit stunted when it comes to social skills, but we can make that work for us. Just tell her exactly what you felt, and how you felt it, and ask if she wants to come over to watch movies and play games, and leave it at that.
Might help to word it in a letter, but don't send it. This stuff is better said face-to-face. It would just be for organizing your thoughts.